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EF 507
QUANTITATIVE METHODS FOR ECONOMICS
AND FINANCE
FALL 2008
Chapter 3
Describing Data: Numerical
Chap 3-1
Describing Data Numerically
Describing Data Numerically
Central Tendency
Variation
Arithmetic Mean
Range
Median
Variance
Mode
Standard Deviation
Coefficient of Variation
Chap 3-2
Measures of Central Tendency
Overview
Central Tendency
Mean
Median
Mode
Midpoint of
ranked values
Most frequently
observed value
n
x
x
i1
i
n
Arithmetic
average
Chap 3-3
Arithmetic Mean

The arithmetic mean (mean) is the most
common measure of central tendency

For a population of N values:
N
x
x1  x 2    x N
μ

N
N
i1
i
Population
values
Population size

For a sample of size n:
n
x
x
i1
n
i
x1  x 2    x n

n
Observed
values
Sample size
Chap 3-4
Arithmetic Mean
(continued)



The most common measure of central tendency
Mean = sum of values divided by the number of values
Affected by extreme values (outliers)
0 1 2 3 4 5 6 7 8 9 10
Mean = 3
1  2  3  4  5 15

3
5
5
0 1 2 3 4 5 6 7 8 9 10
Mean = 4
1  2  3  4  10 20

4
5
5
Chap 3-5
Median

In an ordered list, the median is the “middle”
number (50% above, 50% below)
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Median = 3
Median = 3

Not affected by extreme values
Chap 3-6
Finding the Median

The location of the median:
n 1
Median position 
position in the ordered data
2



If the number of values is odd, the median is the middle number
If the number of values is even, the median is the average of
the two middle numbers
n 1
is not the value of the median, only the
2
position of the median in the ranked data
Note that
Chap 3-7
Mode






A measure of central tendency
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may be no mode
There may be several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode
Chap 3-8
Review Example

Five houses on a hill by the beach
$2,000 K
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
$500 K
$300 K
$100 K
$100 K
Chap 3-9
Review Example:
Summary Statistics
House Prices:
$2,000,000
500,000
300,000
100,000
100,000

Mean:

Median: middle value of ranked data
= $300,000

Mode: most frequent value
= $100,000
Sum 3,000,000
($3,000,000/5)
= $600,000
Chap 3-10
Which measure of location
is the “best”?

Mean is generally used, unless
extreme values (outliers) exist

If outliers exist, then median is often
used, since the median is not
sensitive to extreme values.

Example: Median home prices may be
reported for a region – less sensitive to
outliers
Chap 3-11
Shape of a Distribution

Describes how data are distributed

Measures of shape

Symmetric or skewed
Left-Skewed
Symmetric
Right-Skewed
Mean < Median
Mean = Median
Median < Mean
Chap 3-12
Measures of Variability
Variation
Range

Variance
Standard
Deviation
Coefficient
of Variation
Measures of variation give
information on the spread
or variability of the data
values.
Same center,
different variation
Chap 3-13
Range


Simplest measure of variation
Difference between the largest and the smallest
observations:
Range = Xlargest – Xsmallest
Example:
0 1 2 3 4 5 6 7 8 9 10 11 12
13 14
Range = 14 - 1 = 13
Chap 3-14
Disadvantages of the Range

Ignores the way in which data are distributed
7
8
9
10
11
12
Range = 12 - 7 = 5

7
8
9
10
11
12
Range = 12 - 7 = 5
Sensitive to outliers
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
Range = 5 - 1 = 4
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = 120 - 1 = 119
Chap 3-15
Quartiles

Quartiles split the ranked data into 4 segments with
an equal number of values per segment
25%
Q1



25%
25%
Q2
25%
Q3
The first quartile, Q1, is the value for which 25% of the
observations are smaller and 75% are larger
Q2 is the same as the median (50% are smaller, 50% are
larger)
Only 25% of the observations are greater than the third
quartile
Chap 3-16
Population Variance

Average of squared deviations of values from
the mean (Why “N-1”?)
N

Population variance:
σ 
2
Where
 (x  μ)
i1
2
i
N -1
μ = population mean
N = population size
xi = ith value of the variable x
Chap 3-17
Sample Variance

Average (approximately) of squared deviations
of values from the mean
n

Sample variance:
s 
2
Where
 (x  x)
i1
2
i
n -1
X = arithmetic mean
n = sample size
Xi = ith value of the variable X
Chap 3-18
Population Standard Deviation



Most commonly used measure of variation
Shows variation about the mean
Has the same units as the original data

Population standard deviation:
N
σ
2
(x

μ)
 i
i1
N -1
Chap 3-19
Sample Standard Deviation



Most commonly used measure of variation
Shows variation about the mean
Has the same units as the original data

n
Sample standard deviation:
S
2
(x

x
)
 i
i1
n -1
Chap 3-20
Calculation Example:
Sample Standard Deviation
Sample
Data (xi) :
10
12
14
n=8
s
15
17
18
18
24
Mean = x = 16
(10  X)2  (12  x)2  (14  x)2    (24  x)2
n 1

(10  16)2  (12  16)2  (14  16)2    (24  16)2
8 1

126
7

4.2426
A measure of the “average”
scatter around the mean
Chap 3-21
Measuring variation
Small standard deviation
Large standard deviation
Chap 3-22
Comparing Standard Deviations
Data A
11
12
13
14
15
16
17
18
19
20 21
Mean = 15.5
s = 3.338
20 21
Mean = 15.5
s = 0.926
20 21
Mean = 15.5
s = 4.570
Data B
11
12
13
14
15
16
17
18
19
Data C
11
12
13
14
15
16
17
18
19
Chap 3-23
Advantages of Variance and
Standard Deviation

Each value in the data set is used in the
calculation

Values far from the mean are given extra
weight (because deviations from the mean are
squared)
Chap 3-24
Chebyshev’s Theorem

For any population with mean μ and
standard deviation σ , and k > 1 , the
percentage of observations that fall within
the interval
[μ + kσ]
Is at least
100[1 (1/k )]%
2
Chap 3-25
Chebyshev’s Theorem
(continued)

Regardless of how the data are distributed,
at least (1 - 1/k2) of the values will fall
within k standard deviations of the mean
(for k > 1)

Examples:
At least
within
(1 - 1/12) = 0% ……..... k=1 (μ ± 1σ)
(1 - 1/22) = 75% …........ k=2 (μ ± 2σ)
(1 - 1/32) = 89% ………. k=3 (μ ± 3σ)
Chap 3-26
The Empirical Rule

If the data distribution is bell-shaped, then the
interval:

μ  1σ contains about 68% of the values in
the population or the sample
68%
μ
μ  1σ
Chap 3-27
The Empirical Rule


μ  2σ contains about 95% of the values in
the population or the sample
μ  3σ contains about 99.7% of the values
in the population or the sample
95%
99.7%
μ  2σ
μ  3σ
Chap 3-28
Coefficient of Variation

Measures relative variation

Always in percentage (%)

Shows variation relative to mean

Can be used to compare two or more sets of
data measured in different units
 s
CV     100%
x 
Chap 3-29
Comparing Coefficient
of Variation

Stock A:
 Average price last year = $50
 Standard deviation = $5
s
$5
CVA    100% 
100%  10%
$50
x

Stock B:


Average price last year = $100
Standard deviation = $5
s
$5
CVB    100% 
100%  5%
$100
x
Both stocks
have the same
standard
deviation, but
stock B is less
variable relative
to its price
Chap 3-30
Using Microsoft Excel

Descriptive Statistics can be obtained
from Microsoft® Excel

Use menu choice:
tools / data analysis / descriptive statistics

Enter details in dialog box
Chap 3-31
Using Excel
Use menu choice:

tools / data analysis /
descriptive statistics
Chap 3-32
Using Excel
(continued)

Enter dialog box
details

Check box for
summary statistics

Click OK
Chap 3-33
Excel output
Microsoft Excel
descriptive statistics output,
using the house price data:
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
Chap 3-34
Weighted Mean

The weighted mean of a set of data is
n
w x
x
w
i1


i
i
w 1x1  w 2 x 2    w n x n

 wi
Where wi is the weight of the ith observation
Use when data is already grouped into n classes, with
wi values in the ith class
Chap 3-35
The Sample Covariance

The covariance measures the strength of the linear relationship
between two variables

The population covariance:
N
Cov (x , y)   xy 

 (x  
i
i1
x
)(y i   y )
N
The sample covariance:
n
Cov (x , y)  s xy 


 (x  x)(y  y)
i1
i
i
n 1
Only concerned with the strength of the relationship
No causal effect is implied
Chap 3-36
Interpreting Covariance

Covariance between two variables:
Cov(x,y) > 0
x and y tend to move in the same direction
Cov(x,y) < 0
x and y tend to move in opposite directions
Cov(x,y) = 0
x and y are independent
Chap 3-37
Coefficient of Correlation

Measures the relative strength of the linear
relationship between two variables

Population correlation coefficient (rho):
Cov (x , y)
ρ
σXσY

Sample correlation coefficient (r):
Cov (x , y)
r
sX sY
Chap 3-38
Features of
Correlation Coefficient, r

Unit free (difference with covariance)

Ranges between –1 and 1

The closer to –1, the stronger the negative linear
relationship

The closer to 1, the stronger the positive linear
relationship

The closer to 0, the weaker any positive linear
relationship
Chap 3-39
Scatter Plots of Data with Various
Correlation Coefficients
Y
Y
Y
X
X
r = -1
r = -0.6
Y
r=0
Y
Y
r = +1
X
X
X
r = +0.3
X
r=0
Chap 3-40
Using Excel to Find
the Correlation Coefficient

Select
Tools/Data Analysis

Choose Correlation from
the selection menu

Click OK . . .
Chap 3-41
Using Excel to Find
the Correlation Coefficient
(continued)


Input data range and select
appropriate options
Click OK to get output
Chap 3-42
Interpreting the Result

Scatter Plot of Test Scores
r = 0.733
100


There is a relatively
strong positive linear
relationship between
test score #1
and test score #2
Test #2 Score
95
90
85
80
75
70
70
75
80
85
90
95
100
Test #1 Score
Students who scored high on the first test tended
to score high on second test
Chap 3-43
Obtaining Linear Relationships

An equation can be fit to show the best linear
relationship between two variables:
Y = β0 + β1 X
Where Y is the dependent variable and X is the
independent variable (Ch 12)
Chap 3-44